It is well known that different kinds of expressions of modulus of convexity are essentially based on two geometrical propositions. For one of the propositions we first present a new proof by the Hahn–Banach theorem and intermediate value theorem, then give some corollaries to it which lead to some new expressions of modulus of convexity.

A symmetric random variable is called a Gaussian mixture if it has the same distribution as the product of two independent random variables, one being positive and the other a standard Gaussian random variable. Examples of Gaussian mixtures include random variables with densities proportional to e−|t| p and symmetric p-stable random variables, where p ∈ (0, 2]. We obtain various sharp moment an...

In many cases the convexity of the image of a linear map with range is R n is automatic because of the facial structure of the domain of the map. We develop a four step procedure for proving this kind of " automatic convexity ". To make this procedure more efficient, we prove two new theorems that identify the facial structure of the intersection of a convex set with a subspace in terms of the ...

The purpose of this paper is finding the essential attributes underlying the convexity theorems for momentum maps. It is shown that they are of a topological nature; more specifically, we show that convexity follows if the map is open onto its image and has the so-called local convexity data property. These conditions are satisfied in all the classical convexity theorems and hence they can, in ...

In 1986, Merzbach and Nualart demonstrated a method of transforming a two-parameter point process into a planar Poisson process of unit rate, using random stopping sets. Merzbach and Nualart's theorem applies only to a special class of point processes, since it requires two restrictive conditions: the (F4) condition of conditional independence and the convexity of the 1-compensator. The (F4) co...

M. Brion proved a convexity result for the moment map image of an irreducible subvariety of a compact integral Kähler manifold preserved by the complexification of the Hamiltonian group action. V. Guillemin and R. Sjamaar generalized this result to irreducible subvarieties preserved only by a Borel subgroup. In another direction, L. O’Shea and R. Sjamaar proved a convexity result for the moment...

We study the complexity of finding Tverberg partitions within various convexity spaces. For the classic geometric version of Tverberg’s theorem, we obtain probabilistic algorithms that are weakly polynomial in both the dimension and the number of points. These algorithms extend to other variations, such as the integer version of Tverberg’s theorem. For geodetic convexity on graphs, we show that...

Donaldson conjectured[14] that the space of Kähler metrics is geodesic convex by smooth geodesic and that it is a metric space. Following Donaldson’s program, we verify the second part of Donaldson’s conjecture completely and verify his first part partially. We also prove that the constant scalar curvature metric is unique in each Kähler class if the first Chern class is either strictly negativ...